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23
votes
2answers
782 views

Does the symmetric group on an infinite set have a minimal generating set?

To clarify the terms in the question above: The symmetric group Sym($\Omega$) on a set $\Omega$ consists of all bijections from $\Omega$ to $\Omega$ under composition of functions. A generating set ...
5
votes
1answer
317 views

On Consistency of an Existence

Let $\omega \leq \kappa <2^{\omega}$ , $\omega \leq\lambda \leq \kappa$ and $D(\kappa, \lambda)$ be the statement: For all $ \mathfrak{B} \subseteq \mathbf{P}(\omega)$ with ...
6
votes
1answer
196 views

regularity of ultrafilters

An ultrafilter $U$ is $(\mu,\kappa)$-regular if there is a sequence $\langle X_\alpha : \alpha < \kappa \rangle \subseteq U$ such that for all $y \in [\kappa]^\mu$, $\bigcap_{\alpha \in y} X_\alpha ...
6
votes
1answer
387 views

Order homomorphism functions on $\omega_1$

Let $\omega_1$ be the first uncountable ordinal, same as the set of all countable ordinals. Let $F$ be the set of all functions $f$ from $\omega_1$ minus singleton $0$ into $\omega_1$ that are (a) ...
6
votes
2answers
723 views

If $G \times G \cong H \times H$, then is $G \cong H$? [duplicate]

Let $G,H$ groups. Suppose that $G \times G \cong H \times H$. Then is necessarily $G \cong H$? I know that if $G,H$ are finite groups then it is true (you show that the map $FinGrp \rightarrow ...
6
votes
2answers
335 views

How to show that the chromatic number > aleph_0

Let $V =\{ f | \exists _{\alpha<\omega_1} ( f:\alpha \rightarrow \mathbb{N} \wedge f $ is $ 1-1) \}$. We define $E\subseteq [V]^2 $, such that $\forall_{f,g\in V } (<f,g>\in E ...
8
votes
1answer
204 views

A special c.c.c forcing notion and adding minimal generic reals

This question is related to my question "Forcing with c.c.c forcing notions, Cohen reals and Random reals". A natural way to answer Prikry's conjecture is to build a c.c.c. forcing notion which adds ...
8
votes
1answer
258 views

Weak threads in $\square (\kappa, <\kappa)$ sequences

The following definition is well known ($\kappa$ is regular uncountable cardinal): Definition: a sequence $\mathcal{C} = \langle \mathcal{C}_\alpha | \alpha < \kappa,\,\alpha \text{ limit ...
6
votes
1answer
217 views

Adding large sets not containing countable ground model sets

The question is motivated by Toni's question "Approximation of infinite set in generic extension" (see Approximation of infinite set in generic extension). Before I state the question, let me add ...
5
votes
1answer
137 views

About non-stationary sets of $\omega_1$

Suppose $A$ is a non stationary set of $\omega_1$. Define by induction the following sequence of sets:\ $A_0 = A$ $A_{\alpha+1} = A_{\alpha}'$ [$X'$ is the subset of $X$, of all points the are ...
6
votes
2answers
150 views

Separation of almost disjoint families by ground model almost disjoint families

Suppose that $V$ is a model of $\sf ZFC$, and for concreteness I should point that at this point I am interested in $V=L$ as a ground model. Suppose that $V[c]$ is a Cohen extension of $V$ where $c$ ...
11
votes
1answer
413 views

Are there insane families in $L$?

Let $A,B\subseteq\omega$. We write $A\subseteq^*B$ if $A\setminus B$ is finite, if additionally $B\setminus A$ is infinite then we write $A\subsetneq^*B$, otherwise we write $A=^*B$. We say that a ...
11
votes
0answers
375 views

Hrushovski's Construction

Zilber expressed a conjecture for $\aleph_{1}$- categorical theories.(In 80s) Zilber's Conjecture: The geometry of any $\aleph_{1}$- categorical structure is one of the following: a) Trivial (No ...
6
votes
1answer
645 views

Is there a monster behind the trees?

‎First Fix the following notation:‎ ‎ $‎‎\forall ‎\kappa‎\in Card~~~Tp(‎\kappa‎):="‎\kappa‎~has~tree~property"$‎ ‎‎‎‎ The large cardinals as "monsters of heaven" live everywhere in the land of ...
3
votes
0answers
68 views

Hindman's theorem variant for noncommutative semigroups

The well set proof of Hindman's finite sums theorem applied to noncommutative semigroups yields a sequence of elements such that finite products ordered coherently with this sequence are in one set. ...
6
votes
1answer
227 views

A problem about Ramsey Property

It is known that Ramsey property is a kind of generalizition of pigeon hole principle, and some kinds of Ramsey properties have lots of equivalent forms. We often deal with the case $a\rightarrow ...
7
votes
1answer
229 views

Determinacy from $\omega_1\rightarrow(\omega_1)^{\omega_1}$

Assuming the Axiom of Determinacy (abbreviated AD), Martin showed how to derive a rather strong partition on $\omega_1$, namely that $\omega_1\rightarrow(\omega_1)^{\omega_1}$. In "Infinitary ...
3
votes
1answer
154 views

Countable coloring of a plane

How does one prove existence decomposition of $R^2$ for countable many subsets $A(n)$ s .t.$\forall$ $x,y$ $\epsilon$ $A(n)$ $|x-y|$ is nonrational? I tried thinking of $R^2$ as infinite tree with ...
5
votes
0answers
308 views

Does every strictly increasing, unbounded sequence of positive real numbers contain arbitrarily long, finite subsequences which are “sort of increasing” or “sort of decreasing” (as defined below)?

Is the following true? If $(x_0, x_1, \dots)$ is a strictly increasing, unbounded sequence of positive real numbers, then there exist fixed $M,N \geq 1$ such that the sequence $(x_0, x_1, \dots)$ ...
4
votes
2answers
581 views

Partition calculus question

For $m,n,k < \omega$, consider the equation $X \to (\omega \times k)^{m}_{n}$ What is the smallest $X$ known to satisfy it? Baumgartner-Hajnal theorem gives a satisfactory answer for $m=2$, but ...
7
votes
2answers
751 views

Size of stationary sets

What can we say about the size of stationary subsets of $P_{\kappa}(\lambda)$ for infinite cardinals $\kappa, \lambda,$ especially when $\kappa=\aleph_1.$ Please give me some references, if there are ...
24
votes
5answers
1k views

Game involving 'asking questions about a real'

Consider the following game, played by two players, called Q and A, in a time frame t = 1, 2, .... At every time point i, Q mentions some $Q_i \subset \mathbb{R}$, after which A mentions $A_i$ such ...
10
votes
0answers
224 views

Combinatorial Hilbert spaces

Any closed subspace $V\subset {\ell}^2(\omega)$ has associated to it a subset ${\cal S}_V$ of ${\cal P}(\omega)$, call it a combinatorial Hilbert space, namely the set of all supports of all vectors ...
7
votes
2answers
686 views

failure of $\square(\kappa)$ at an inaccessible $\kappa$

How can we force the failure of $\square(\kappa)$ at an inaccessible $\kappa$, where $\square(\kappa)$ is defined as follows: There is a sequence $(C_i:i< \kappa)$ such that: (1) $C_{i+1} = ...